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Proving monad-law-1
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Sun, 30 Nov 2014 22:26:50 +0900
parents e95f15af3f8b
children 1f4ea5cb153d
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5ba82f107a95 Define Similar in Agda
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1 open import list
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6e6d646d7722 Split basic functions to file
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2 open import basic
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e0ba1bf564dd Apply level to some functions
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e0ba1bf564dd Apply level to some functions
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4 open import Level
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742e62fc63e4 Define Monad-law 1-4
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5 open import Relation.Binary.PropositionalEquality
742e62fc63e4 Define Monad-law 1-4
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6 open ≡-Reasoning
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8 module delta where
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11 data Delta {l : Level} (A : Set l) : (Set (suc l)) where
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12 mono : A -> Delta A
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13 delta : A -> Delta A -> Delta A
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14
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15 deltaAppend : {l : Level} {A : Set l} -> Delta A -> Delta A -> Delta A
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16 deltaAppend (mono x) d = delta x d
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17 deltaAppend (delta x d) ds = delta x (deltaAppend d ds)
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18
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19 headDelta : {l : Level} {A : Set l} -> Delta A -> A
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20 headDelta (mono x) = x
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21 headDelta (delta x _) = x
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23 tailDelta : {l : Level} {A : Set l} -> Delta A -> Delta A
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24 tailDelta (mono x) = mono x
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25 tailDelta (delta _ d) = d
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5ba82f107a95 Define Similar in Agda
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6ce83b2c9e59 Proof Functor-laws
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27
6ce83b2c9e59 Proof Functor-laws
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28 -- Functor
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29 fmap : {l ll : Level} {A : Set l} {B : Set ll} -> (A -> B) -> (Delta A) -> (Delta B)
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30 fmap f (mono x) = mono (f x)
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31 fmap f (delta x d) = delta (f x) (fmap f d)
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35 -- Monad (Category)
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36 eta : {l : Level} {A : Set l} -> A -> Delta A
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37 eta x = mono x
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742e62fc63e4 Define Monad-law 1-4
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38
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39 bind : {l ll : Level} {A : Set l} {B : Set ll} -> (Delta A) -> (A -> Delta B) -> Delta B
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40 bind (mono x) f = f x
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41 bind (delta x d) f = delta (headDelta (f x)) (bind d (tailDelta ∙ f))
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46b15f368905 Define bind and mu for Infinite Delta
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46b15f368905 Define bind and mu for Infinite Delta
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43 mu : {l : Level} {A : Set l} -> Delta (Delta A) -> Delta A
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44 mu d = bind d id
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45
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46 returnS : {l : Level} {A : Set l} -> A -> Delta A
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47 returnS x = mono x
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49 returnSS : {l : Level} {A : Set l} -> A -> A -> Delta A
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50 returnSS x y = deltaAppend (returnS x) (returnS y)
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0bc402f970b3 Proof Monad-law 1
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53 -- Monad (Haskell)
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54 return : {l : Level} {A : Set l} -> A -> Delta A
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55 return = eta
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23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
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57 _>>=_ : {l ll : Level} {A : Set l} {B : Set ll} ->
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58 (x : Delta A) -> (f : A -> (Delta B)) -> (Delta B)
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59 (mono x) >>= f = f x
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60 (delta x d) >>= f = delta (headDelta (f x)) (d >>= (tailDelta ∙ f))
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6ce83b2c9e59 Proof Functor-laws
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64 -- proofs
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66 -- Functor-laws
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68 -- Functor-law-1 : T(id) = id'
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69 functor-law-1 : {l : Level} {A : Set l} -> (d : Delta A) -> (fmap id) d ≡ id d
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70 functor-law-1 (mono x) = refl
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71 functor-law-1 (delta x d) = cong (delta x) (functor-law-1 d)
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72
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73 -- Functor-law-2 : T(f . g) = T(f) . T(g)
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74 functor-law-2 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
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75 (f : B -> C) -> (g : A -> B) -> (d : Delta A) ->
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76 (fmap (f ∙ g)) d ≡ ((fmap f) ∙ (fmap g)) d
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77 functor-law-2 f g (mono x) = refl
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78 functor-law-2 f g (delta x d) = cong (delta (f (g x))) (functor-law-2 f g d)
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81 -- Monad-laws (Category)
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84 data Int : Set where
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85 O : Int
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86 S : Int -> Int
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87
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88 _+_ : Int -> Int -> Int
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89 O + n = n
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90 (S m) + n = S (m + n)
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91
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92 n-tail : {l : Level} {A : Set l} -> Int -> ((Delta A) -> (Delta A))
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93 n-tail O = id
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94 n-tail (S n) = tailDelta ∙ (n-tail n)
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95
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96 flip : {l : Level} {A : Set l} -> (f : A -> A) -> f ∙ (f ∙ f) ≡ (f ∙ f) ∙ f
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97 flip f = refl
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98
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99 n-tail-plus : {l : Level} {A : Set l} -> (n : Int) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n)
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100 n-tail-plus O = refl
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101 n-tail-plus (S n) = begin
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102 n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩
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103 (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩
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104 tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩
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105 n-tail (S (S n))
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106 ∎
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107
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108 postulate n-tail-add : {l : Level} {A : Set l} -> (n m : Int) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m)
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109 postulate int-add-assoc : (n m : Int) -> n + m ≡ m + n
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110 postulate int-add-right-zero : (n : Int) -> n ≡ n + O
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111 postulate int-add-right : (n m : Int) -> S n + S m ≡ S (S (n + m))
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112
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113
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
114
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
115
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
116
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
117
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
118 tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Int) -> (x : A) ->
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
119 (n-tail n) (mono x) ≡ (mono x)
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
120 tail-delta-to-mono O x = refl
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
121 tail-delta-to-mono (S n) x = begin
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
122 n-tail (S n) (mono x) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
123 tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
124 tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
125 tailDelta (mono x) ≡⟨ refl ⟩
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
126 mono x
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
127 ∎
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
128
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
129 monad-law-1-5 : {l : Level} {A : Set l} -> (m : Int) (n : Int) -> (ds : Delta (Delta A)) ->
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
130 n-tail n (bind ds (n-tail m)) ≡ bind (n-tail n ds) (n-tail (m + n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
131 monad-law-1-5 O O ds = refl
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
132 monad-law-1-5 O (S n) (mono ds) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
133 n-tail (S n) (bind (mono ds) (n-tail O)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
134 n-tail (S n) ds ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
135 bind (mono ds) (n-tail (S n)) ≡⟨ cong (\de -> bind de (n-tail (S n))) (sym (tail-delta-to-mono (S n) ds)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
136 bind (n-tail (S n) (mono ds)) (n-tail (S n)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
137 bind (n-tail (S n) (mono ds)) (n-tail (O + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
138 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
139 monad-law-1-5 O (S n) (delta d ds) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
140 n-tail (S n) (bind (delta d ds) (n-tail O)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
141 n-tail (S n) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta ))) (sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
142 ((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta )) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
143 (n-tail n) (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
144 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
145 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S n))) (n-tail-plus n) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
146 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
147 bind (n-tail (S n) (delta d ds)) (n-tail (O + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
148 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
149 monad-law-1-5 (S m) n (mono (mono x)) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
150 n-tail n (bind (mono (mono x)) (n-tail (S m))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
151 n-tail n (n-tail (S m) (mono x)) ≡⟨ cong (\de -> n-tail n de) (tail-delta-to-mono (S m) x)⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
152 n-tail n (mono x) ≡⟨ tail-delta-to-mono n x ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
153 mono x ≡⟨ sym (tail-delta-to-mono (S m + n) x) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
154 (n-tail (S m + n)) (mono x) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
155 bind (mono (mono x)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (mono x))) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
156 bind (n-tail n (mono (mono x))) (n-tail (S m + n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
157 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
158 monad-law-1-5 (S m) n (mono (delta x ds)) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
159 n-tail n (bind (mono (delta x ds)) (n-tail (S m))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
160 n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
161 n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
162 n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add n m) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
163 n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
164 n-tail (m + n) ds ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
165 ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
166 n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
167 n-tail (S m + n) (delta x ds) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
168 bind (mono (delta x ds)) (n-tail (S m + n)) ≡⟨ cong (\de -> bind de (n-tail (S m + n))) (sym (tail-delta-to-mono n (delta x ds))) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
169 bind (n-tail n (mono (delta x ds))) (n-tail (S m + n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
170 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
171 monad-law-1-5 (S m) O (delta d ds) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
172 n-tail O (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
173 (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
174 delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
175 bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
176 bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (int-add-right-zero (S m)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
177 bind (n-tail O (delta d ds)) (n-tail (S m + O))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
178 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
179 monad-law-1-5 (S m) (S n) (delta d ds) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
180 n-tail (S n) (bind (delta d ds) (n-tail (S m))) ≡⟨ cong (\t -> t ((bind (delta d ds) (n-tail (S m))))) (sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
181 ((n-tail n) ∙ tailDelta) (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
182 ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
183 (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
184 (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
185 bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (int-add-right m n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
186 bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
187 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
188 bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
189 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
190
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
191 monad-law-1-4 : {l : Level} {A : Set l} -> (n : Int) -> (dd : Delta (Delta A)) ->
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
192 headDelta ((n-tail n) (bind dd tailDelta)) ≡ headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
193 monad-law-1-4 O (mono dd) = refl
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
194 monad-law-1-4 O (delta dd dd₁) = refl
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
195 monad-law-1-4 (S n) (mono dd) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
196 headDelta (n-tail (S n) (bind (mono dd) tailDelta)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
197 headDelta (n-tail (S n) (tailDelta dd)) ≡⟨ cong (\t -> headDelta (t dd)) (n-tail-plus (S n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
198 headDelta (n-tail (S (S n)) dd) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
199 headDelta (n-tail (S (S n)) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S (S n)) (headDelta de))) (sym (tail-delta-to-mono (S n) dd)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
200 headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (mono dd))))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
201 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
202 monad-law-1-4 (S n) (delta d ds) = begin
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
203 headDelta (n-tail (S n) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
204 headDelta (n-tail (S n) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta)))) ≡⟨ cong (\t -> headDelta (t (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta))))) (sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
205 headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
206 headDelta (n-tail n (bind ds (tailDelta ∙ tailDelta))) ≡⟨ {!!} ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
207 headDelta (n-tail (S (S n)) (headDelta ((n-tail n ds)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
208 headDelta (n-tail (S (S n)) (headDelta ((n-tail n ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S (S n)) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
209 headDelta (n-tail (S (S n)) (headDelta (n-tail (S n) (delta d ds))))
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
210 ∎
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
211
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
212 monad-law-1-2 : {l : Level} {A : Set l} -> (d : Delta (Delta A)) -> headDelta (mu d) ≡ (headDelta (headDelta d))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
213 monad-law-1-2 (mono _) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
214 monad-law-1-2 (delta _ _) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
215
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
216 monad-law-1-3 : {l : Level} {A : Set l} -> (n : Int) -> (d : Delta (Delta (Delta A))) ->
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
217 bind (fmap mu d) (n-tail n) ≡ bind (bind d (n-tail n)) (n-tail n)
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
218 monad-law-1-3 O (mono d) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
219 monad-law-1-3 O (delta d ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
220 bind (fmap mu (delta d ds)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
221 bind (delta (mu d) (fmap mu ds)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
222 delta (headDelta (mu d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta dx (bind (fmap mu ds) tailDelta)) (monad-law-1-2 d) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
223 delta (headDelta (headDelta d)) (bind (fmap mu ds) tailDelta) ≡⟨ cong (\dx -> delta (headDelta (headDelta d)) dx) (monad-law-1-3 (S O) ds) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
224 delta (headDelta (headDelta d)) (bind (bind ds tailDelta) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
225 bind (delta (headDelta d) (bind ds tailDelta)) (n-tail O) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
226 bind (bind (delta d ds) (n-tail O)) (n-tail O)
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
227 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
228 monad-law-1-3 (S n) (mono (mono d)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
229 bind (fmap mu (mono (mono d))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
230 bind (mono d) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
231 (n-tail (S n)) d ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
232 bind (mono d) (n-tail (S n)) ≡⟨ cong (\t -> bind t (n-tail (S n))) (sym (tail-delta-to-mono (S n) d))⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
233 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
234 bind (n-tail (S n) (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
235 bind (bind (mono (mono d)) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
236 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
237 monad-law-1-3 (S n) (mono (delta d ds)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
238 bind (fmap mu (mono (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
239 bind (mono (mu (delta d ds))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
240 n-tail (S n) (mu (delta d ds)) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
241 n-tail (S n) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ cong (\t -> t (delta (headDelta d) (bind ds tailDelta))) (sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
242 (n-tail n ∙ tailDelta) (delta (headDelta d) (bind ds tailDelta)) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
243 n-tail n (bind ds tailDelta) ≡⟨ monad-law-1-5 (S O) n ds ⟩
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
244 bind (n-tail n ds) (n-tail (S n)) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
245 bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S n)) ≡⟨ cong (\t -> (bind (t (delta d ds)) (n-tail (S n)))) (n-tail-plus n) ⟩
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
246 bind (n-tail (S n) (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
247 bind (bind (mono (delta d ds)) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
248 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
249 monad-law-1-3 (S n) (delta (mono d) ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
250 bind (fmap mu (delta (mono d) ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
251 bind (delta (mu (mono d)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
252 bind (delta d (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
253 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
254 delta (headDelta ((n-tail (S n)) d)) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) d)) de) (monad-law-1-3 (S (S n)) ds) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
255 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
256 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
257 delta (headDelta ((n-tail (S n)) d)) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
258 delta (headDelta ((n-tail (S n)) (headDelta (mono d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail (S n)) (headDelta de))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (sym (tail-delta-to-mono (S n) d)) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
259 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (mono d))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
260 bind (delta (headDelta ((n-tail (S n)) (mono d))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
261 bind (bind (delta (mono d) ds) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
262 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
263 monad-law-1-3 (S n) (delta (delta d dd) ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
264 bind (fmap mu (delta (delta d dd) ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
265 bind (delta (mu (delta d dd)) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
266 delta (headDelta ((n-tail (S n)) (mu (delta d dd)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
267 delta (headDelta ((n-tail (S n)) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta (t (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))))(sym (n-tail-plus n)) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
268 delta (headDelta (((n-tail n) ∙ tailDelta) (delta (headDelta d) (bind dd tailDelta)))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
269 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
270 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta (headDelta ((n-tail n) (bind dd tailDelta))) de) (monad-law-1-3 (S (S n)) ds) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
271 delta (headDelta ((n-tail n) (bind dd tailDelta))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ cong (\de -> delta de ( (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))))) (monad-law-1-4 n dd) ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
272 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
273 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
274 delta (headDelta ((n-tail (S n)) (headDelta (n-tail n dd)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
73
0ad0ae7a3cbe Proving monad-law-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 72
diff changeset
275 delta (headDelta ((n-tail (S n)) (headDelta (((n-tail n) ∙ tailDelta) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ cong (\t -> delta (headDelta ((n-tail (S n)) (headDelta (t (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n))))) (n-tail-plus n) ⟩
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
276 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) (delta d dd))))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
277 bind (delta (headDelta ((n-tail (S n)) (delta d dd))) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
278 bind (bind (delta (delta d dd) ds) (n-tail (S n))) (n-tail (S n))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
279 ∎
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
280
71
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
281 {-
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
282 monad-law-1-3 (S n) (mono d) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
283 bind (fmap mu (mono d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
284 bind (mono (mu d)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
285 n-tail (S n) (mu d) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
286 bind (n-tail (S n) d) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
287 bind (bind (mono d) (n-tail (S n))) (n-tail (S n))
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
288 ∎
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
289 monad-law-1-3 (S n) (delta d ds) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
290 bind (fmap mu (delta d ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
291 bind (delta (mu d) (fmap mu ds)) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
292 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
293 delta (headDelta ((n-tail (S n)) (mu d))) (bind (fmap mu ds) (n-tail (S (S n)))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
294 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (n-tail (S (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
295 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (n-tail (S (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
296 delta (headDelta ((n-tail (S n)) (headDelta ((n-tail (S n)) d)))) (bind (bind ds (tailDelta ∙ (n-tail (S n)))) (tailDelta ∙ (n-tail (S n)))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
297 bind (delta (headDelta ((n-tail (S n)) d)) (bind ds (tailDelta ∙ (n-tail (S n))))) (n-tail (S n)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
298 bind (bind (delta d ds) (n-tail (S n))) (n-tail (S n))
71
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
299 ∎
56da62d57c95 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 70
diff changeset
300 -}
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
301
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
302 -- monad-law-1 : join . fmap join = join . join
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
303 monad-law-1 : {l : Level} {A : Set l} -> (d : Delta (Delta (Delta A))) -> ((mu ∙ (fmap mu)) d) ≡ ((mu ∙ mu) d)
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
304 monad-law-1 (mono d) = refl
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
305 {-
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
306 monad-law-1 (delta x (mono d)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
307 (mu ∙ fmap mu) (delta x (mono d)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
308 mu (fmap mu (delta x (mono d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
309 mu (delta (mu x) (mono (mu d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
310 delta (headDelta (mu x)) (bind (mono (mu d)) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
311 delta (headDelta (mu x)) (tailDelta (mu d)) ≡⟨ cong (\dx -> delta dx (tailDelta (mu d))) (monad-law-1-2 x) ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
312 delta (headDelta (headDelta x)) (tailDelta (mu d)) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
313 delta (headDelta (headDelta x)) (bind (tailDelta d) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
314 mu (delta (headDelta x) (tailDelta d)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
315 mu (delta (headDelta x) (bind (mono d) tailDelta)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
316 mu (mu (delta x (mono d))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
317 (mu ∙ mu) (delta x (mono d))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
318 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
319 monad-law-1 (delta x (delta d ds)) = begin
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
320 (mu ∙ fmap mu) (delta x (delta d ds)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
321 mu (fmap mu (delta x (delta d ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
322 mu (delta (mu x) (delta (mu d) (fmap mu ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
323 delta (headDelta (mu x)) (bind (delta (mu d) (fmap mu ds)) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
324 delta (headDelta (mu x)) (delta (headDelta (tailDelta (mu d))) (bind (fmap mu ds) (tailDelta ∙ tailDelta))) ≡⟨ {!!} ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
325 delta (headDelta (headDelta x)) (delta (headDelta (tailDelta (headDelta (tailDelta d)))) (bind (bind ds (tailDelta ∙ tailDelta)) (tailDelta ∙ tailDelta))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
326 delta (headDelta (headDelta x)) (bind (delta (headDelta (tailDelta d)) (bind ds (tailDelta ∙ tailDelta))) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
327 delta (headDelta (headDelta x)) (bind (bind (delta d ds) tailDelta) tailDelta) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
328 mu (delta (headDelta x) (bind (delta d ds) tailDelta)) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
329 mu (mu (delta x (delta d ds))) ≡⟨ refl ⟩
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
330 (mu ∙ mu) (delta x (delta d ds))
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
331 ∎
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
332 -}
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
333
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
334 monad-law-1 (delta x d) = begin
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
335 (mu ∙ fmap mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
336 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
337 mu (fmap mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
338 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
339 mu (delta (mu x) (fmap mu d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
340 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
341 delta (headDelta (mu x)) (bind (fmap mu d) tailDelta)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
342 ≡⟨ cong (\x -> delta x (bind (fmap mu d) tailDelta)) (monad-law-1-2 x) ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
343 delta (headDelta (headDelta x)) (bind (fmap mu d) tailDelta)
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
344 ≡⟨ cong (\d -> delta (headDelta (headDelta x)) d) (monad-law-1-3 (S O) d) ⟩
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
345 delta (headDelta (headDelta x)) (bind (bind d tailDelta) tailDelta)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
346 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
347 mu (delta (headDelta x) (bind d tailDelta))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
348 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
349 mu (mu (delta x d))
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
350 ≡⟨ refl ⟩
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
351 (mu ∙ mu) (delta x d)
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
352 ∎
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
353
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
354
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
355
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
356 {-
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
357 -- monad-law-2 : join . fmap return = join . return = id
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
358 -- monad-law-2-1 join . fmap return = join . return
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
359 monad-law-2-1 : {l : Level} {A : Set l} -> (d : Delta A) ->
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
360 (mu ∙ fmap eta) d ≡ (mu ∙ eta) d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
361 monad-law-2-1 (mono x) = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
362 monad-law-2-1 (delta x d) = {!!}
63
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
363
474ed34e4f02 proving monad-law-1 ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 62
diff changeset
364
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
365 -- monad-law-2-2 : join . return = id
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
366 monad-law-2-2 : {l : Level} {A : Set l } -> (d : Delta A) -> (mu ∙ eta) d ≡ id d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
367 monad-law-2-2 d = refl
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
368
35
c5cdbedc68ad Proof Monad-law-2-2
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 34
diff changeset
369
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
370 -- monad-law-3 : return . f = fmap f . return
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
371 monad-law-3 : {l : Level} {A B : Set l} (f : A -> B) (x : A) -> (eta ∙ f) x ≡ (fmap f ∙ eta) x
36
169ec60fcd36 Proof Monad-law-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 35
diff changeset
372 monad-law-3 f x = refl
27
742e62fc63e4 Define Monad-law 1-4
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 26
diff changeset
373
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
374
39
b9b26b470cc2 Add Comments
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 38
diff changeset
375 -- monad-law-4 : join . fmap (fmap f) = fmap f . join
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
376 monad-law-4 : {l ll : Level} {A : Set l} {B : Set ll} (f : A -> B) (d : Delta (Delta A)) ->
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
377 (mu ∙ fmap (fmap f)) d ≡ (fmap f ∙ mu) d
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
378 monad-law-4 f d = {!!}
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
379
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
380
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
381
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
382
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
383 -- Monad-laws (Haskell)
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
384 -- monad-law-h-1 : return a >>= k = k a
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
385 monad-law-h-1 : {l ll : Level} {A : Set l} {B : Set ll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
386 (a : A) -> (k : A -> (Delta B)) ->
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
387 (return a >>= k) ≡ (k a)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
388 monad-law-h-1 a k = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
389
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
390
40
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
391
a7cd7740f33e Add Haskell style Monad-laws and Proof Monad-laws-h-1
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 39
diff changeset
392 -- monad-law-h-2 : m >>= return = m
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
393 monad-law-h-2 : {l : Level}{A : Set l} -> (m : Delta A) -> (m >>= return) ≡ m
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
394 monad-law-h-2 (mono x) = refl
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
395 monad-law-h-2 (delta x d) = cong (delta x) (monad-law-h-2 d)
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
396
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
397
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
398 -- monad-law-h-3 : m >>= (\x -> k x >>= h) = (m >>= k) >>= h
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
399 monad-law-h-3 : {l ll lll : Level} {A : Set l} {B : Set ll} {C : Set lll} ->
43
90b171e3a73e Rename to Delta from Similar
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 42
diff changeset
400 (m : Delta A) -> (k : A -> (Delta B)) -> (h : B -> (Delta C)) ->
41
23474bf242c6 Proof monad-law-h-2, trying monad-law-h-3
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 40
diff changeset
401 (m >>= (\x -> k x >>= h)) ≡ ((m >>= k) >>= h)
59
46b15f368905 Define bind and mu for Infinite Delta
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 57
diff changeset
402 monad-law-h-3 (mono x) k h = refl
70
18a20a14c4b2 Change prove method. use Int ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 69
diff changeset
403 monad-law-h-3 (delta x d) k h = {!!}
69
295e8ed39c0c Change headDelta definition. return non-delta value
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 68
diff changeset
404
72
e95f15af3f8b Trying prove infinite-delta. but I think this definition was missed.
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 71
diff changeset
405 -}